Graphs of Linear Equations - §3.3
Fall 2013 - Math 1010
y = mx + b
(y − y1 ) = m(x − x1 )
Ax + By = C
(Math 1010)
M 1010 §3.3
1 / 10
Roadmap
I
Slopes of linear equations.
I
Graphing a linear equation.
I
Parallel and perpendicular lines.
I
Graphs of absolute value equations.
(Math 1010)
M 1010 §3.3
2 / 10
Linear Equations
A linear equation in two variables has an equation with two variables, x
and y , raised to the first power each. Their use in mathematical models is
deep. For example: y = 10.7296x + 33.4744
(Math 1010)
M 1010 §3.3
3 / 10
Linear Equations
y = 3.45x − 87.52
(Math 1010)
M 1010 §3.3
4 / 10
Slope
The slope between a pair of points (x1 , y1 ) and (x2 , y2 ) is
m=
y2 − y1
.
x2 − x1
This can be called, ”the-change-in-y over the-change-in-x,” also,
”rise-over-run.”
(Math 1010)
M 1010 §3.3
5 / 10
Slope
The slope between a pair of points (x1 , y1 ) and (x2 , y2 ) is
m=
y2 − y1
.
x2 − x1
This can be called, ”the-change-in-y over the-change-in-x,” also,
”rise-over-run.”
3
2
1
0
-0
1
2
3
Slope
The slope between a pair of points (x1 , y1 ) and (x2 , y2 ) is
m=
y2 − y1
.
x2 − x1
This can be called, ”the-change-in-y over the-change-in-x,” also,
”rise-over-run.”
3
2
Rise
1
0
-0
1
2
3
Slope
The slope between a pair of points (x1 , y1 ) and (x2 , y2 ) is
m=
y2 − y1
.
x2 − x1
This can be called, ”the-change-in-y over the-change-in-x,” also,
”rise-over-run.”
3
2
Rise
1
Run
0
-0
(Math 1010)
1
2
M 1010 §3.3
3
5 / 10
Forms of a Line
Horizontal lines have a slope of m = 0. Then the y −intercept, (0, b),
matters. The line has the form y = b.
Vertical lines have an undefined slope. The x−intercept, (a, 0), matters,
and the equation of such a line is x = a.
Otherwise, lines may have the forms:
Algebraic Form
y = mx + b
Slope-Intercept
(y − y1 ) = m(x − x1 )
Ax + By = C
(Math 1010)
Name of the Form
Point-Slope
Standard Form
M 1010 §3.3
6 / 10
Point-slope Form
A linear equation of the form
y = mx + b
is in slope-intercept form, and has a slope value m, and y −intercept (0, b).
(Math 1010)
M 1010 §3.3
7 / 10
Point-slope Form
A linear equation of the form
y = mx + b
is in slope-intercept form, and has a slope value m, and y −intercept (0, b).
Example: The equation 3x − 2y = −10 has slope-intercept form
(Math 1010)
M 1010 §3.3
7 / 10
Point-slope Form
A linear equation of the form
y = mx + b
is in slope-intercept form, and has a slope value m, and y −intercept (0, b).
Example: The equation 3x − 2y = −10 has slope-intercept form
3
y = x + 5.
2
(Math 1010)
M 1010 §3.3
7 / 10
Point-slope Form
A linear equation of the form
y = mx + b
is in slope-intercept form, and has a slope value m, and y −intercept (0, b).
Example: The equation 3x − 2y = −10 has slope-intercept form
3
y = x + 5.
2
The slope is
3
2
and y −intercept is (0, 5).
Remember: A pair of points make a line. With this, you can use the
slope-formula. After, you need to slope and one point to find the
y −intercept.
(Math 1010)
M 1010 §3.3
7 / 10
Graphing A Line
Using the slope and intercept as graphing aids, we can graph
2
y = x +1
3
Plot the y −intercept (0, 1).
Locate a second point by moving up 2 units and moving right 3 units.
3
2
1
0
-0
1
2
3
2
3
Graphing A Line
Using the slope and intercept as graphing aids, we can graph
2
y = x +1
3
Plot the y −intercept (0, 1).
Locate a second point by moving up 2 units and moving right 3 units.
3
2
1•
0
-0
1
2
3
2
3
Graphing A Line
Using the slope and intercept as graphing aids, we can graph
2
y = x +1
3
Plot the y −intercept (0, 1).
Locate a second point by moving up 2 units and moving right 3 units.
•
3
2
2
1•
3
0
-0
1
2
3
2
3
Graphing A Line
Using the slope and intercept as graphing aids, we can graph
2
y = x +1
3
Plot the y −intercept (0, 1).
Locate a second point by moving up 2 units and moving right 3 units.
2
3
•
3
2
2
1•
3
0
-0
(Math 1010)
1
2
M 1010 §3.3
3
8 / 10
Parallel and Perpendicular lines
Equations of lines with the same slope are parallel.
Example: The lines y = x + 28 and y = x + 17 both have a slope of 1.
Equations of lines are perpendicular when their slopes are opposite and
reciprocals.
Example: The line y = 73 x − 3 has a slope of 37 .
The line y = − 73 x + 9 has a slope of − 37 .
(Math 1010)
M 1010 §3.3
9 / 10
Assignment
Assignment:
For Monday:
1. Exercises from §3.3 due Monday, September 23.
2. Exercises from $ 3.2 may be turned in Monday, September 23.
3. Read section 3.4.
(Math 1010)
M 1010 §3.3
10 / 10